where km is a positive constant depending only on m. It follows almost immediately from this theorem that the average order of A(n) is \pi2n/6 log n. Let A^*(n) = sumi = 1rp1. Then the average order of A^*(n) is also \pi2n/6 log n, and the average order of A(n)-A^*(n) is log log n. For any fixed positive integer M, the set of solutions to A(n)-A^*(n) = M has a positive natural density. Now A(n) = n if and only if n is a prime or n = 4. Call n a ''special number'' if n\equiv O(mod A(n)) and n\neq A(n), and let {ln} be the sequence of special numbers. This paper's first author has previously proved that the sequence {ln} is infinite [Srinivasa Ramanujan Commemoration Volume, Oxford Press, Madras, India, (1974) part II] . Denote by L(x) the number of \elln \leq x. It is shown that there exist positive constants c, c' such that